Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 9 (2013), 055, 17 pages      arXiv:1302.3727      https://doi.org/10.3842/SIGMA.2013.055

$\mathfrak{spo}(2|2)$-Equivariant Quantizations on the Supercircle $S^{1|2}$

Najla Mellouli a, Aboubacar Nibirantiza b and Fabian Radoux b
a) University of Sfax, Higher Institute of Biotechnology, Route de la Soukra km 4, B.P. no 1175, 3038 Sfax, Tunisia
b) University of Liège, Institute of Mathematics, Grande Traverse, 12 - B37, B-4000 Liège, Belgium

Received February 18, 2013, in final form August 15, 2013; Published online August 23, 2013

Abstract
We consider the space of differential operators $\mathcal{D}_{\lambda\mu}$ acting between $\lambda$- and $\mu$-densities defined on $S^{1|2}$ endowed with its standard contact structure. This contact structure allows one to define a filtration on $\mathcal{D}_{\lambda\mu}$ which is finer than the classical one, obtained by writting a differential operator in terms of the partial derivatives with respect to the different coordinates. The space $\mathcal{D}_{\lambda\mu}$ and the associated graded space of symbols $\mathcal{S}_{\delta}$ ($\delta=\mu-\lambda$) can be considered as $\mathfrak{spo}(2|2)$-modules, where $\mathfrak{spo}(2|2)$ is the Lie superalgebra of contact projective vector fields on $S^{1|2}$. We show in this paper that there is a unique isomorphism of $\mathfrak{spo}(2|2)$-modules between $\mathcal{S}_{\delta}$ and $\mathcal{D}_{\lambda\mu}$ that preserves the principal symbol (i.e. an $\mathfrak{spo}(2|2)$-equivariant quantization) for some values of $\delta$ called non-critical values. Moreover, we give an explicit formula for this isomorphism, extending in this way the results of [Mellouli N., SIGMA 5 (2009), 111, 11 pages] which were established for second-order differential operators. The method used here to build the $\mathfrak{spo}(2|2)$-equivariant quantization is the same as the one used in [Mathonet P., Radoux F., Lett. Math. Phys. 98 (2011), 311-331] to prove the existence of a $\mathfrak{pgl}(p+1|q)$-equivariant quantization on $\mathbb{R}^{p|q}$.

Key words: equivariant quantization; supergeometry; contact geometry; orthosymplectic Lie superalgebra.

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